Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 257
Such a transformation is called an infinitesimal transformation. The expression.f 3f af
Uf- el, + + I....( + f.(4)
- ax, ax ax.................................
is adopted as its symbol, since Uf. t is the increment assigned to any function
f (x,..., xn) by the infinitesimal transformation.
2. If the transformations of the continuous ensemble (1) are so related that the
successive application of any two of them is equivalent to a transformation belonging
to the same family, (1) is called a continuous group of oo transformations.
Let the family (1) be a continuous group; suppose further that the group contains
the inverse transformation of every transformation in it: that is, that the resolution
of the equations (1) with regard to x,,..., x,, gives a system of the form
X1=Xi (1,..., Xni, b), X2 = X2 (X',..., n b),...xn = Xn (x,../,... x,,', b)... (),
where b is a constant depending only on a.
Under these conditions it is easy to see that the group contains an infinitesimal
transformation; for, if T, is the transformation of the group corresponding to the
parameter value a, the inverse Ta~- of T, is also found in the group. Further the
transformation Ta+sa corresponding to the parameter value a + Sa, is the transformation
of the group differing infinitesimally from Ta. The product Ta+s Ta-' which, by the
assumed group property, belongs to the group, differs then infinitesimally from the
transformation T7a Ta-'; but the latter is the identical transformation; thus the group
contains a transformation possessed of the properties attributed to an infinitesimal
transformation in the preceding paragraph.
3. Conversely, every infinitesimal transformation is contained in a determinate
continuous group. This may be made clear in the following manner. The given infinitesimal transformation assigns the infinitesimal increments
1= 1(1,... )t,., îx=*.(, Zn) t,.................... (6)
to the variables x,,..., x,n, on neglecting infinitely small quantities of a higher order;
if t be interpreted as the time, x,..., x, as point-coordinates in a space of n dinmensions, ît as a time increment, and îx1,..., Sx, as the corresponding increments of
i,..., xn, then the equations (6) determine a stationary flow in space of n dimensions.
After an interval of time t the point (x,,..., x,) will have assuined the new position
(X/,..., xn); the latter position will be obtained by integrating the simultaneous
system
dx - dx,' dx,' ddn
(L, *, x,., X/) "' - n (1/..3..,X d)
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